Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r. To find the maximum volume of a rectangular box inscribed in a sphere of radius r, we need to find the dimensions of the box that will give us the greatest volume.

We can start by finding the volume of a sphere with a radius of r. This will be our starting point. From there, we can begin to adjust the dimensions of the box until we find the combination that gives us the maximum volume.

The formula for finding the volume of a sphere is:

V = 4/3 * π * r^3

Using this formula, we can plug in different values for r until we find the combination that gives us the maximum volume.

For example, if r = 1, then the volume of the sphere is:

V = 4/3 * π * 1^3

V = 4/3 * π * 1

V = 4.1887902047903

If we increase r to 2, the volume of the sphere becomes:

V = 4/3 * π * 2^3

V = 4/3 * π * 8

V = 33.510321638291

As you can see, increasing the sphere’s radius increases its volume. This means that to find the maximum volume for our box, we need to make sure that the sphere’s radius is as large as possible.

The dimensions of the box that will give us the maximum volume are:

length = 2r

width = 2r

height = 2r

Using these dimensions, we can plug them into the formula for finding the volume of a rectangular box. This formula is:

V = l*w*h

where l is the length, w is the width, and he is the height.

Plugging in our dimensions, we get:

V = (2r)*(2r)*(2r)

V = 8r^3

Now that we have the volume of our box, we can plug in different values for r until we find the combination that gives us the maximum volume.

For example, if r = 1, then the volume of the box is:

V = 8 * 1^3

V = 8 * 1

V = 8

If we increase r to 2, the volume of the box becomes:

V = 8 * 2^3

V = 8 * 8

V = 64

As you can see, increasing the sphere’s radius increases the volume of our box. This means that to find the maximum volume for our box, we need to make sure that the sphere’s radius is as large as possible.